Parsing does only a syntax check. If you had a statement like "exec dbo.ThisStoredProcedureDoesNotExist", and clicked on parse, it will parse with no errors. So successful parsing by itself doesn't tell you anything.

First thing to check is whether the stored procedure indeed exists in the database. In SSMS object explorer, expand the stored procedures node under the database name -> Programmability and make sure it is there. If it is there, then you need permission to execute it. You won't be able to grant yourself permission, so you will need to login using a higher priviledged login and do "grant execute on StoredProcName to YourUsername"